Let
Now, we take
Once we have the length of the projection, the dot product is:
There is another way:
BECAUSE, imagine if we use a tasformation with only one dimension for 2D vectors.
Now, with this transformation, the span of the system becomes a line, because both unit vectors are on the same dimension.
So if we put any vector through this transformation, it will go along this line.
Since the components of our vector are adding value to the same dimension, for us to obtain our output vector, we just need to sum its components:
So the new vector/scalar is -2.
And we can see that applying the transformation to the vector is the same operation that we described before.
In the first way, we are taking the norm/magnitudes of each vector and comparing them in one dimension.
In the second way we are collapsing everything in one single dimension(the one of the first vector), then the other vector will automatically project onto the dimension.
TO BE REVISED!
BECAUSE, the dot product lets you see if two vectors are similar.
In fact, if two vectors are generally going in the same direction, the dot product will be positive and big.
On the contrary, if the vectors are dissimilar, the dot product will be smaller.
If the two vectors are perpendicular, the
If the two vectors are going in opposite directions, the dot product will be negative.
Order doesn't matter, we can take the projection of either one of the two vectors and the dot product will stay the same.
Works only if you want to compare vectors that are in the same vector (dimensional) space.
So, the inner product is essentially a way of measuring the similarity or correlation between two vectors. If the vectors are similar, meaning their corresponding components are aligned in a similar direction, the inner product will be a large positive value. If the vectors are dissimilar, meaning their corresponding components are pointing in different directions or canceling each other out, the inner product will be smaller, possibly negative.